It seems they hang out in villages to try to recruit troops, so they're waiting there for more troops to respawn. I had a companion party finally get enough troops and they started hunting looters around a city.
No, it is not necessary to have your own kingdom. You simply must form an army (bottom right icon) and add the parties of your clan that will join without cost if they can
After forming a "Party" and giving them troops to support them, they set off on their own expeditions; I guess attempting to increase their skills and influence like we are. (COOL!) That companion was then captured by a leader of a minor faction, broke free, and was found in a city tavern. (The 'log' keeps you informed up until the breakout of their last location, so far.)
EDIT: SCOUTING is also important for this as well simply because you may venture past important figures simply because you can't see them. Once they are out of view, may have trouble locating them
The encyclopedia helps keep tabs on 'individuals' last location; Something tracking down rulers has taught me!
No, it is not necessary to have your own kingdom. You simply must form an army (bottom right icon) and add the parties of your clan that will join without cost if they can